Lines, Central & Inscribed Angles
Inscribed Quadrilaterals
Inscribed
Triangles
Triangles
Sectors and Arc Lengths
Degrees and
Radians
Radians
100
Name both the central angle and the inscribed angle.
Central: angle WCX
Inscribed: angle WVX
100
True or false: All squares and rectangles are cyclic quadrilaterals.
Yes, this is true because we can create a circle from all 4 vertices of a square and a rectangle. Additionally, the diagonal angles of both squares and rectangles add up to 180º.
100
Fill in the blanks:
The circumcenter of a triangle is the intersection of the three _____________ of the triangle's _________.
The incenter of a triangle is the intersection of the three _____________ of the triangle's _________.
1. Perpendicular bisectors
2. Sides
3. Angle bisectors
4. Angles
100
Calculate the area of a sector with a central angle of 40º and a radius of 12cm.
40º/360º = 1/9
1/9*144π = 16πcm squared
100
What is the measure of the angle that makes an entire circle in radians?
2π radians
200
Fill in the blanks:
A central angle has a vertex on the _________ of a circle and is made of two _______.
An inscribed angle has a vertex on the _________ of a circle and is made of two _______.
1. center point
2. radii
3. circumference
4. chords
200
Draw your own cyclic quadrilateral that is not a square or rectangle and label the possible angles.
Answers may vary!
200
True or False: An isosceles triangle has a circumcenter and an incenter in the same location.
False, an equilateral triangle has a circumcenter and an incenter in the same location.
200
A circle has radius 10 centimeters. Suppose an arc on the circle has length 8π centimeters. What is the measure of the central angle whose radii define the arc?
144º
200
Find the radius of a circle with a central angle of 5 radians and an arc length of 20 inches.
radians = arc length/radius
5 rad = 20 in/radius
radius = 4 inches
300
True or False & Explain: All diameters are chords, but a radius is only sometimes a chord.
All diameters are chords because they go from one end of the circle to the other, but a radius can never be a chord because it never touches both ends to the circumference of the circle since one end must be at the center of the circle.
300
Quadrilateral ABCD is cyclic. What is the value of angles C and D?
Angle A = 54º
Angle B = 82º
Angle C = 126º
Angle D = 98º
300
Where is the circumcenter located on a triangle with the angles 24º, 103º, and 53º?
It is located outside the triangle because it is an obtuse triangle.
300
Find the arc length of a sector with angle 80º and a radius of 5 cm.
80º/360º = 2/9*2*5*π = 2/9*10π = 20/9*π = 2.22222π cm = about 6.98 cm
300
Find the area of a sector with a central angle of 2π/3 radians and a radius of 6 inches.
2π/3 * 1/2π = 1/3
36π * 1/3 = 12π inches squared
400
Arc AB measures 124º, what is the measurement of angle ADB and angle ACB?
BOTH ADB and ACB have a measurement of 62º because they are corresponding to the same arc. No matter how the vertex of an inscribed angle moves around the circle, the measure of the angle stays the same.
400
Find the measure of arc JML.
360º-90º=270º
400
What are the measures of line segments OI, OJ, and OH?
They are all 5 units long because OH uses 12, 13, and 5 as a Pythagorean triple and all three segments are the same length since they are all radii of the inscribed circle within the triangle.
400
A circle has a radius of 20 centimeters and a central angle that measures 216 degrees. Find the length of the arc defined by this central angle.
216º/360º = 0.6
40π * 0.6 = 24 π cm
400
What is the value of 5π/8 radians in degrees?
5π/8 * 1/π * 180º = 5/8 * 180º = 112.5º
500
The measure of angle CAB is 33º. What is the measure of angle x?
Angle x is 57º because arc BC is 2*33º, so 66º. Arc AC is 180º, so 180º-66º= 114º. To find angle x, we can divide 114º by 2 and we get 57º.
500
Find the measure of angles X, Y, and Z.
X = (106º+58º)/2 = 164º/2 = 82º
y = 180º-82º = 98º
Z = 180º-93º = 87º
500
The distance from B to O measures 12 inches. Find the area of the inscribed circle inside triangle ABC.
Since triangle BDO is a 30º, 60º, 90º triangle and the side across from 90º is 12 inches, the side DO must be 6 inches. The area of the circle is therefore 36π inches squared.
500
Find the area of the shaded sector.
148º/360º = 0.41111...
7.5^2 π yd^2 = 56.25π yd^2
23.125π yd^2
500
Write all of the angles for the entire unit circle from memory in BOTH degrees and radians!
